Question

Two rods, one of Al and other of steel, having initial lengths $l_1$ and $l_2$ are connected together to form a single rod of length $l_1+l_2$. The coefficient of linear expansion of aluminium and steel are ${\alpha }_A$ and ${\alpha }_S$ respectively. If the length of each rod increases by same amount when the temperature is raised by $t^0$C then find the relation $\frac{l_1}{l_1+l_2}$.

• A. $\frac{{\alpha }_A}{{\alpha }_A+{\alpha }_S}$
• B. $\frac{{\alpha }_S}{{\alpha }_A}$
• C. $\frac{{\alpha }_A}{{\alpha }_S}$
• D. $\frac{{\alpha }_S}{{\alpha }_A+{\alpha }_S}$

Answer 1

The correct option is D $\frac{{\alpha }_S}{{\alpha }_A+{\alpha }_S}$

length is inversely proportional to coefficient of linear expansion, thus
$\frac{l_2}{l_1}=\frac{{\alpha }_A}{{\alpha }_S}$
or $1+\frac{l_2}{l_1}=1+\frac{{\alpha }_A}{{\alpha }_S}$
or
$\frac{l_1+l_2}{l_1}=\frac{{\alpha }_A+{\alpha }_S}{{\alpha }_S}$
or
$\frac{l_1}{l_1+l_2}=\frac{{\alpha }_S}{{\alpha }_A+{\alpha }_S}$